Problem 46 - Algebra and Trigonometry

Question

a. Use the Leading Coefficient Test to determine the graph's end behavior. b. Find the \(x\)-intercepts. State whether the graph crosses the \(x\)-axis, or touches the \(x\) -axis and turns around, at each intercept. c. Find the \(y\)-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. \(f(x)=-x^{4}+4 x^{2}\)

Step-by-Step Solution

Answer

End behavior: As \(x->\infty\) or \(x->-\infty\), \(f(x)->-\infty\). X-intercepts at \(x = -2, 0, 2\) where the graph crosses the x-axis. Y-intercept at \(y = 0\). The graph is symmetric about the y-axis and not symmetric about the origin. The graph should have at most 3 turning points.

Step 1: Determining the End Behavior

The end behavior of a graph is determined by the degree and the leading coefficient of the polynomial. For the function \(f(x)=-x^{4}+4 x^{2}\), the degree is 4 (even) and the leading coefficient is -1. An even degree with a negative leading coefficient means both ends of the graph point downwards. So as \(x\) approaches infinity (\(x->\infty\)) or negative infinity (\(x->-\infty\)), \(f(x)\) approaches negative infinity (\(f(x)->-\infty\)).

Step 2: Finding the x-intercepts

To find the x-intercepts, set \(f(x) = 0\) and solve for \(x\). So, solving \(-x^{4}+4 x^{2} = 0\) we have to factor the equation first. Factoring gives \(x^{2}(4 - x^{2}) = 0\). Setting each factor equal to zero gives \(x = 0\) and \(x = ±2\). Thus, the graph crosses the x-axis at \(x = 0\) and \(x = ±2\).

Step 3: Finding the y-intercept

The y-intercept of a function is the value of \(f(x)\) when \(x = 0\). For the function, substituting \(x = 0\) gives \(f(0) = -0^{4}+4*0^{2} = 0\). Thus, the y-intercept is \(y = 0\).

Step 4: Determining Symmetry

A function is symmetric about the y-axis if \(f(-x) = f(x)\) and symmetric about the origin if \(f(-x) = -f(x)\). Substituting \(x\) for \(-x\) in the function gives \(-(-x)^{4}+4 (-x)^{2} = -x^{4}+4 x^{2}\), which is the same as the original function. Therefore, the function is symmetric about the y-axis. Additionally, it's not symmetric about the origin since \(f(-x) ≠ -f(x)\).

Step 5: Graphing the Function

With the x-intercepts at \(-2, 0, 2\), the y-intercept at \(0\), and knowing the end behavior and symmetry, we graph the function. To check the correctness of the graph, note that a polynomial of degree \(n = 4\) can have up to \(n - 1 = 3\) turning points. Our graph should have at most 3 turning points.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a handy tool for predicting the end behavior of polynomial graphs—how the graph behaves as it heads towards positive or negative infinity. When dealing with this concept, focus on the leading term of the polynomial, which is the term with the highest power of the variable.

To apply the test, look at the degree of the polynomial and the sign of its leading coefficient. If the degree is even and the leading coefficient is positive, both ends of the graph will point upward. Conversely, if the leading coefficient is negative, both ends will point downward. For polynomials with odd degrees, if the leading coefficient is positive, the end behavior will be up on the right and down on the left; if negative, vice versa.

The function f(x) = -x^4 + 4x^2 illustrates an instance where the degree is 4 (even) and the leading coefficient is -1 (negative), telling us that both tails of the graph point down, leading to an intuitive visualization of the graph's behavior at extreme values of x.

X-intercepts of a Polynomial

X-intercepts, also known as zeroes or roots, are the points where the graph of a polynomial crosses or touches the x-axis. These occur when the value of f(x) is zero. To find the x-intercepts, you'll need to solve the equation f(x) = 0.

In the example f(x) = -x^4 + 4x^2, factoring helps us identify the intercepts. Once factored into x^2(4 - x^2) = 0, we set each factor to zero and solve for x, providing us with x = 0, x = +2, and x = -2 as intercepts. Noting whether the graph crosses or touches the x-axis at these points helps us map the function more accurately.

Y-intercept of a Graph

The y-intercept is found where the graph crosses the y-axis, which is when x = 0. To locate the y-intercept of a function, you simply substitute 0 for x in the equation and calculate f(0).

For instance, with our function f(x) = -x^4 + 4x^2, plugging in 0 gives f(0) = 0, indicating that the y-intercept is at the origin, (0,0). This information is crucial for graphing as it provides a starting point on the graph.

Symmetry in Algebra

Symmetry in algebra refers to a geometric balance in a graph. A function can be symmetric about the y-axis or about the origin. If you replace x with -x in the function and the function remains unchanged, it's y-axis symmetric. If replacing x with -x results in the original function being multiplied by -1, it's symmetric about the origin.

For the function f(x) = -x^4 + 4x^2, symmetry can be tested by substituting -x for x. Since this substitution does not change the function, it is symmetric about the y-axis. This characteristic impacts how the graph is drawn, ensuring that for every point on one side of the y-axis, there is a mirror point on the opposite side.

Turning Points of Polynomial Functions

Turning points are places where the graph changes direction, from increasing to decreasing or vice versa. The number of turning points in a polynomial function is at most the degree of the polynomial minus one. However, just because a polynomial can have that many turning points doesn't mean it will. It’s the maximum possible.

Looking again at f(x) = -x^4 + 4x^2, which is a fourth-degree polynomial (n = 4), it may have up to 3 (n - 1) turning points. When graphing, if the number of observed turning points exceeds this limit, it could mean the graph has been plotted incorrectly, and errors in plotting or calculating should be checked.